Mathematics > Numerical Analysis

Title:Analysis of the ensemble Kalman filter for inverse problems

Abstract: The ensemble Kalman filter (EnKF) is a widely used methodology for state
estimation in partial, noisily observed dynamical systems, and for parameter
estimation in inverse problems. Despite its widespread use in the geophysical
sciences, and its gradual adoption in many other areas of application, analysis
of the method is in its infancy. Furthermore, much of the existing analysis
deals with the large ensemble limit, far from the regime in which the method is
typically used. The goal of this paper is to analyze the method when applied to
inverse problems with fixed ensemble size. A continuous-time limit is derived
and the long-time behavior of the resulting dynamical system is studied. Most
of the rigorous analysis is confined to the linear forward problem, where we
demonstrate that the continuous time limit of the EnKF corresponds to a set of
gradient flows for the data misfit in each ensemble member, coupled through a
common pre-conditioner which is the empirical covariance matrix of the
ensemble. Numerical results demonstrate that the conclusions of the analysis
extend beyond the linear inverse problem setting. Numerical experiments are
also given which demonstrate the benefits of various extensions of the basic
methodology.